DAMTP 2002 - 31

BORN-INFELD KINEMATICS

Frederic P Schuller

Department of Applied Mathematics and Theoretical PhysicsCentre for Mathematical Sciences

University of Cambridge

Wilberforce RoadCambridge CB3 0WA, U.K.

F.P.Schuller@damtp.cam.ac.uk

15 January 2002

ABSTRACT.

Born-Infeld theory possesses particular dynamical symmetries due to its maximal fieldstrength, leading to a maximal acceleration for a particle coupled to it. These symme-tries can be encoded in a pseudo-complex structure on the tangent bundle of spacetime,which we study in both the flat and the curved case.Pseudo-complexification of Minkowski spacetime and the Lorentz group give a maximal ac-celaration extension of special relativity, in complete agreement with special relativity fornon-accelerated observers. Pseudo-complexification of relativistic Lagrangians leads to ex-tended relativistic dynamics, as previously studied in the literature. The assumption of afinite upper bound on accelerations is shown to be equivalent to a non-commutative geom-etry on spacetime.In the case of curved spacetime, the Tachibana-Okumura no-go theorem singles out pseudo-complex manifolds as those compatible with a maximal acceleration and the strong principleof equivalence. Thus we can lift the Einstein field equations and give the maximal accelera-tion extension of general relativity as a tangent bundle theory, in complete agreement withgeneral relativity in the absence of other forces.The dynamics of a free massive particle with maximal acceleration is studied using pseudo-complex techniques. A consistent pseudo-complex Lorentz invariant coupling to Born-Infeldelectrodynamics is constructed via the lifting of the Kaluza-Klein metric to the tangentbundle.We close with a remark on the appearance of anti-commutation relations upon quantisation.

Contents

1 Introduction 3

2 Dynamics Goes Kinematics 5

3 Pseudo-Complex Modules 73.1 Ring of Pseudo-Complex Numbers . . . . . . . . . . . . . . . . . . . 73.2 P-Modules and Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . 93.3 The Pseudo-Complex Lorentz Group . . . . . . . . . . . . . . . . . . 103.4 The Pseudo-Complex Sequence . . . . . . . . . . . . . . . . . . . . . 10

4 Maximal Acceleration Extension of Special Relativity 124.1 Mathematical Structure . . . . . . . . . . . . . . . . . . . . . . . . . 124.2 Linear Uniform Acceleration in Extended Special Relativity . . . . 144.3 Special Transformations in SOP(1, 3) . . . . . . . . . . . . . . . . . . 154.4 Physical Postulates of Extended Special Relativity . . . . . . . . . . 184.5 Born-Infeld Theory Revisited . . . . . . . . . . . . . . . . . . . . . . 19

5 Pseudo-Complex Manifolds 205.1 Lifts to the Tangent Bundle . . . . . . . . . . . . . . . . . . . . . . . 215.2 Adapted Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.3 Lifts of the Spacetime Metric . . . . . . . . . . . . . . . . . . . . . . 235.4 Connections on the Tangent Bundle . . . . . . . . . . . . . . . . . . 235.5 Orbidesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.6 Orbidesic Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . 265.7 The Tachibana-Okumura No-Go Theorem . . . . . . . . . . . . . . . 26

6 Maximal Acceleration Extension of General Relativity 286.1 Physical Postulates of Extended General Relativity . . . . . . . . . . 286.2 Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

7 Point Particle Dynamics 307.1 Free Massive Particles . . . . . . . . . . . . . . . . . . . . . . . . . . 307.2 Kaluza-Klein induced coupling to Born-Infeld theory . . . . . . . . . 33

8 Quantisation 36

9 Conclusion 38

2

1 Introduction

Over the last two decades, there has been some interest in and speculation on theexistence of a finite upper bound on accelerations in fundamental physics, motivatedfrom results in quantum field theory [2] and string theory [3]. Both of these theoriesbuild on special relativity as a kinematical framework, but upon quantization, afinite upper bound on accelerations apparently enters through the back door.There are interesting results on the promise that the inclusion of a maximal acceler-ation on the classical level already, will positively modify the convergence behaviourof loop diagrams in quantum field theory [1]. These calculations, however, use an ad-hoc introduction of a maximal acceleration and the authors point out that a rigorouscheck would require a consistent classical framework which their approach is lacking.

In this paper, we present such a maximal acceleration extension of special and gen-eral relativity, obtained by ’kinematization’ of dynamical symmetries of the Born-Infeld action, as explained in the next section.

This leads to a non-trivial lift of special and general relativity to the tangent bun-dle, or equivalently, the cotangent bundle, of spacetime. Born [4], among others,remarked that in contrast to our deep understanding of non-relativistic mechanicsin the Hamiltonian formulation on phase space, special and general relativity areformulated on spacetime only, and the structure of the associated phase space is onlypoorly understood and thus little used. Clearly, this is likewise true for all dynamicaltheories based on the kinematical framework of special and general relativity, mostnotably quantum field theory and string theory. In view of the importance of thenon-relativistic phase space structure for the transition to quantum theory, Bornregards a phase space formulation of general relativity as a necessary step towardsa reconciliation of gravity and quantum mechanics on a descriptional level.Following the observation of Caianiello [12] that the (co-)tangent bundle of spacetimemight be an appropriate stage for a maximal acceleration extension of special rela-tivity, several attempts have been made to equip the tangent bundle with a complexstructure, both in the case of flat [7] and curved [6] spacetime. These approachesare all inconsistent, as we will show in section 5.7 that a complex structure is incom-patible with the metric geometry needed to impose an upper bound on accelerations.

Anticipating this negative result, we develop the theory of pseudo-complex modulesand later generalize to pseudo-complex manifolds. This is then demonstrated toprovide the appropriate phase space geometry circumventing the above mentionedno-go theorem. We obtain a lift of the Einstein field equations to the tangent bun-dle, thus enabling us to formulate a theory of gravity with finite upper bound onaccelerations due to non-gravitational forces. Spacetime concepts are regarded asderived ones, and indeed the existence of a finite maximal acceleration is seen togive rise to a non-commutative geometry on spacetime, which becomes commutativeagain in the limit of infinite maximal acceleration.

3

In dynamical theories featuring a maximal acceleration, second order derivativesseemed an inconvenient but necessary evil [14, 15] in order to dynamically enforcethe maximal acceleration. Exploiting pseudo-complexification techniques, however,we achieve to recast Lagrangians with second order derivatives into first order form.Rather than being just a notational trick, the dynamical information on the maxi-mal acceleration is absorbed into the extended kinematics. This is shown in detailin section 7.

Putting the full machinery for generally curved pseudo-complex manifolds to workallows a rigorous discussion of a Kaluza-Klein induced coupling of a submaximallyaccelerated particle to Born-Infeld theory.

The last chapter presents a thought on the implications of the pseudo-complex phasespacetime structure for the transition to quantum theory. One lesson there is thatthe pseudo-complex structure embraces the complex structure, rather than being inopposition to it.

4

2 Dynamics Goes Kinematics

Prior to Einstein’s formulation of special relativity, the (at the time) peculiar role ofthe speed of light in electrodynamics was regarded a consequence of the particulardynamics given by Maxwell theory,

LMaxwell = −14FµνF

µν . (1)

In modern parlance, one would say that the boost-invariance of (1) was consideredto be a dynamical symmetry of this particular theory, not necessarily present inother fundamental theories of nature.Einstein’s idea, however, was to regard the symmetries of Maxwell theory as kine-matical symmetries, i.e. due to the structure of the underlying spacetime, and hencenecessarily present in any theory acting on this stage. Convinced of the correctnessof special relativity, we think today that all sensible dynamics must be Poincarecovariant.Born-Infeld electrodynamics

LBI = det (η + bF )12 , (2)

although indistinguishable from Maxwell theory at large1 distances, modifies itsshort range behaviour essentially. In particular, Born-Infeld theory features a max-imal electric field strength

Emax = b−1, (3)

controlled by the parameter b. Hence, coupling a massive particle of charge e toBorn-Infeld theory,

L = LBI + Lfree particle + Lint, (4)

where

Lfree particle = pµxµ − 12λ

(pµp

µ −m2), (5)

Lint = −exµAµ, (6)

we see from the equations of motion, that such a particle can at most experience anacceleration

a =eb−1

m. (7)

Clearly, this is a feature solely due to the particular dynamics of the model (4). Butone may well ask whether one can redo Einstein’s trick and convert the dynamicalfeature of a maximal acceleration into a kinematical one. Taking the resulting

1the scale given by the parameter b

5

framework seriously, viable dynamics are then required to be covariant under theinduced symmetry group, which will turn out to include the Lorentz group.In order to make an educated guess of how the kinematization of the Born-Infeldsymmetries could be achieved, consider the Born-Infeld Lagrangian

L = det1/2(ηµν +

e

maFµν

), (8)

where the parameter b is fixed according to relation (7). It is fairly easy to verifythat this can be written

det1/4 (ηµν) det1/4(ηµν +

e2

m2a2FµρFν

ρ

)= det1/4

(emaFµν −ηµνηµν − e

maFµν

)︸ ︷︷ ︸

≡Hmn

, (9)

introducing the matrix H and using the determinant identity for n × n matricesA,N,M ∣∣∣∣ −At N

M A

∣∣∣∣ = (−1)n |N | ∣∣M +AN−1At∣∣ . (10)

Let x be the worldline of a particle of mass m in spacetime, and u its four-velocity.Denoting the corresponding curve in ’velocity phase spacetime’

Y m ≡ (axµ, uµ) , (11)

we recognize that

Hmn =im

a[Y m, Y n] (12)

corresponds to the canonical commutation relations in presence of an electromag-netic field

[xµ, xν ] = −iem−2a−2Fµν , (13)[xµ, pν ] = −iηµν , (14)[pµ, pν ] = ieFµν , (15)

with an additional a−2-suppressed coordinate non-commutativity, also controlled bythe electromagnetic field strength tensor. Hence, assuming this highly suppressednon-commutativity, we have the surprising result

LBI = det1/4 ([Y m, Y n]) , (16)

suggesting an encoding of the dynamical symmetries of Born-Infeld theory in anappropriate geometry of the tangent or cotangent bundle2 of Minkowski spacetime.

2we can freely switch between those two as we are given a metric on spacetime

6

Conventionally, the complex structure of the cotangent bundle is the key to a ge-ometrical understanding of Hamiltonian systems [5], and lead to the commutationrelations in the transition to bosonic quantum theory. However, in section 5.7, weshow that unless the underlying spacetime is flat, a complex structure of the associ-ated phase space is incompatible with a finite upper bound on accelerations. Hence,even the slightest perturbation of Minkowski spacetime renders such approaches[6, 7] inconsistent, and hence we deem them unphysical at all.Fortunately, there is a way round this negative result in form of equipping phasespacetime with a pseudo-complex structure, which naturally leads to anticommu-tation relations for the associated quantum theory, as shown in section 8. The keymathematical ingredient is the ring of pseudo-complex numbers, and the moduleand algebra structures built upon them. The following chapter is devoted to thesemathematical developments.

3 Pseudo-Complex Modules

This section introduces the concept of pseudo-complex numbers, explores some prop-erties and then focuses on the somewhat subtle pseudo-complexification of real vec-torspaces and Lie algebras.

3.1 Ring of Pseudo-Complex Numbers

The pseudo-complex ring is the set

P ≡ a+ Ib | a, b ∈ R , (17)

equipped with addition and multiplication laws induced by those on R, where I /∈ R

is a pseudo-complex structure, i.e. I2 = 1. There is a matrix representation

1 ≡(

1 00 1

), I ≡

(0 11 0

), (18)

such that addition and multiplication on P are given by matrix addition and multi-plication. It is easily verified that P is a commutative unit ring with zero divisors

P0± ≡ λ(1± I) |λ ∈ R . (19)

P0+,P

0− /P are the only multiplicative ideals in P, thus they are both maximal ideals.Hence the only fields one can construct from P are

P/P0+

∼= P0− ∼= R,

P/P0− ∼= P

0+∼= R, (20)

which are too trivial for our purposes. Hence we have to stick to P and deal withits ring structure.For p = a+ Ib ∈ P, define the pseudo-complex conjugate

p ≡ a− Ib. (21)

7

The map

| · |2 : P −→ R

|p|2 ≡ pp = a2 − b2 (22)

is a semi-modulus on the ring P, which now decomposes into three classes

P = P+ ∪ P

− ∪ P0, (23)

according to the sign of the modulus:

P+ ≡

p ∈ P : |p|2 > 0,

P− ≡

p ∈ P : |p|2 < 0, (24)

P0 ≡

p ∈ P : |p|2 = 0

= P0±.

Define the exponential map

exp : P −→ P+

exp(p) ≡∞∑n=0

pn

n!. (25)

In terms of p = a+ Ib this yields

exp(a+ Ib) = exp(a) cosh(b) + I sinh(b) (26)

and hence exp converges on all of P and is one-to-one. As P is commutative, thefunctional identity

exp(p) exp(q) = exp(p+ q) (27)

holds for all pseudo-complex numbers p, q ∈ P. Note, in particular, that for anyexp(p), there is always a unique multiplicative inverse, namely exp(−p).Using the exponential map, we get the ’polar’ representations

P+ = r exp(Iψ)|r, ψ ∈ R ,

P− = Ir exp(Iψ)|r, ψ ∈ R , (28)

P0± = λ(1 ± I)|λ ∈ R .

It is easily verified that the symmetry transformations on P preserving the semimodulus are the (1 + 1)-dimensional Lorentz group:

OP(1) ∼= OR(1, 1) (29)

8

3.2 P-Modules and Lie Algebras

Usually, the Lie algebras occuring in physical applications are real or complex vectorspaces. However, the most general algebraic definition of a Lie algebra only demandsit be a module over a commutative ring. Hence we can sensibly define the pseudo-complex extension LP of a real Lie algebra L by

LP ≡ t+ Is|t, s ∈ L . (30)

which is a free3P-module, as L is a vector space. The Lie bracket on LP is induced by

that on L. Its multilinearity follows directly from the commutativity of P. Clearly,

dimP(LP) = dimR(L) =12

dimR(LP).

Let Ti with i = 1, . . . ,dimR(L) be the generators of the real Lie algebra L. Then wehave

LP = 〈Ti〉P = 〈Ti, ITi〉R , (31)

and we will switch between these two pictures where appropriate. As P is commu-tative, we have

[exp(piTi)]−1 = exp(−piTi). (32)

Hence we can obtain the connection component Gid of the associated Lie group Gby exponentiation of the algebra

Gid = exp (PL) = exp (RLP) . (33)

Let the real vector space V be a representation of the real Lie group L. Then LP

acts naturally on the pseudo-complexification of V ,

VP ≡ x+ Iu|x, u ∈ V (34)

which is a free P-module of dimension dimR V . VP also being an R-vectorspace ofdimension 2 dimR, we can identify VP with the tangent bundle TV via

VP = x+ Iu|x ∈ V, u ∈ TxV . (35)

The bundle projection is then given in this language by

π : TV ≡ VP −→ V

π(X) ≡ 12(X + X) (36)

3i.e. having a basis

9

3.3 The Pseudo-Complex Lorentz Group

Let η ≡ diag (1,−1, . . . ,−1) have signature (1, n − 1) and consider the pseudo-complex extension soP(1, n− 1) of the real Lorentz algebra soR(1, n− 1). Exponen-tiation gives the pseudo-complex Lorentz group

SOP(1, n − 1) ≡ Λ ∈ Mat (n,P) |ΛT ηΛ = η, detΛ = 1

(37)

Clearly, for any U ∈ TPn, the expression

(U,U) ≡ UµUνηµν (38)

is invariant under the action of SOP(1, n−1). Expanding (38) using Uµ = uµ+ Iaµ,u, a ∈ R

r+s yields

(U,U) = (uµuµ + aµaµ) + I (2uµaµ) . (39)

Let Mµν be the standard generators of the real Lorentz-group, then

soP(1, n − 1) = 〈Mµν , IMµν〉R

(40)

and the pseudo-complex linear combinations

Gµν ≡Mµν + IMµν , Gµν = Mµν − IMµν (41)

generate two decoupled real Lorentz algebras, and hence

soP(1, n − 1) ∼= soR(1, n − 1)⊕ soR(1, n − 1). (42)

Note that the real and pseudoimaginary part of (39) are preserved separately underthe action of OP(1, n − 1). Hence, we can switch between the picture of a metricmodule and a bimetric vector space:

(Pn, η) ∼= (TR

n, ηD, ηH), (43)

where ηD = η ⊗ 1 and ηH = η ⊗ I denote the diagonal and horizontal lifts to thetangent bundle [8], respectively, of the Minkowski metric.

3.4 The Pseudo-Complex Sequence

Define inductively, for all n ∈ N,

P(0) ≡ R, (44)

P(n+1) ≡

1⊗ a+ I ⊗ b | a, b ∈ P

(n), (45)

the sequence of pseudo-complex rings. P(n) is called the pseudo-complex ring of rank

n. Commutativity is easily shown by induction. Clearly, P(n) is a vector space of

dimension 2n, with canonical basis

1⊗ 1⊗ · · · ⊗ 1, 1 ⊗ 1⊗ · · · ⊗ I, . . . , I ⊗ I ⊗ · · · ⊗ I.

10

Identifying the b-th canonical basis vector b in P(n) with a binary number 0 6 b 6

2n − 1 via 1 = 0, I = l, multiplication between basis elements corresponds to the’exlcusive or’ operation t

t 0 l

0 0 ll l 0

(46)

Hence the multiplication of two n-th rank pseudo-complex numbers

p =2n−1∑b=0

pb b, q =2n−1∑b=0

qc c

is given by

pq =2n−1∑b,c

pbqc b t c

=2n−1∑b,d

pbqbtd d, (47)

observing that b t c = d if and only if b t d = c. We use the extension of this resultto the infinite-dimensional case to define the multiplication law on

P(∞) ≡ p : N −→ R , (48)

the set of all real sequences.From the binary representation, the sets P

(n) and P(∞) are seen to be commutative

rings with unit 0. Hence on can define the n-th rank pseudo-complexification of areal Lie algebra L

LP(n) ≡

2n−1∑b=0

tbb | tb ∈ L

(49)

which is a free module of dimension

dim(n)P

(LP(n)) = dimR(L) = 2−n dimR(L

P(n)), (50)

and the n-th rank pseudo-complexification of a real vectorspace V ,

VP(n) ≡

2n−1∑b=0

xbb |xb ∈ V

(51)

being a free P(n)-module, or a real vectorspace of dimension 2n dimR(V ). We can

identify VP(n) with the n-th tangent bundle T nV , i.e. the bundle of n-jets [8].

11

4 Maximal Acceleration Extension of Special Relativity

We obtain a maximal acceleration extension of special relativity by pseudo-complexificationof Minkowski spacetime

R1,3 −→ P

1,3 (52)

and appropriate lifts of all spacetime concepts to the resulting module. Alternativelyto pseudo-complexification, all this can be understood in terms of lifts to the tangentbundle of spacetime, at the cost of having to deal with two metrics.That the geometry obtained in this way indeed encodes the Born-Infeld kinematics,is shown in section 4.5. The theory is formulated entirely independent of spacetimeconcepts. This involves the slight paradigm shift of thinking of the objects definedon pseudo-complexified spacetime as primary, rather than being induced from thefamiliar spacetime concepts, which will be regarded as derived ones.

4.1 Mathematical Structure

We introduce a fundamental constant a of dimension length−1, called maximal accel-eration. The stage for extended relativistic physics is pseudo-complexified Minkowskispacetime (

P4, η

) ≡ P1,3,

equipped with a pseudo-complex valued two-form

η : P4 × P

4 −→ P (53)

of signature (1, 3). Note that without further restriction, η is not a real-valued semi-norm on P

1,3, as its pseudo-imaginary part is non-vanishing in general.The symmetry algebra of P

1,3 is, by construction, the pseudo-complexified Lorentzalgebra

oP(1, 3).

We introduce a preferred class of coordinate systems, generalizing the inertial framesof special relativity. A basis

e(µ)

of TP

1,3 with

η(e(µ), e(ν)

)= ηµν ≡ diag (1,−1,−1,−1) (54)

is called a uniform basis or uniform frame. Coordinates of P1,3 given with respect

to such a basis are called uniform coordinates.We will always work in uniform coordinates in this chapter. Clearly, under the ac-tion of SOP(1, 3), a uniform basis is transformed to a uniform basis.

Let x : R −→ R1,3 be a timelike curve in real Minkowski spacetime. Then the

natural lift X ≡ x∗ to pseudo-complexified spacetime

x : R −→ R1,3

↓ ∗X : R −→ P

1,3 (55)

12

is defined by

X ≡ x∗ ≡ ax+ Iu (56)

where u ≡ dxdτ and dτ2 ≡ η(dx, dx). Let X : R −→ P

1,3 be a curve in configurationspace. X is called an orbit iff

X = π(X)∗, (57)

where ∗ denotes the natural lift (56) and π the projection (36).The line element of the projection π(X) of an arbitrary orbit X in a particularuniform frame is denoted by dτ and given by

dτ2 ≡ η (dπ(X), dπ(X)) . (58)

Note that this quantity is frame-dependent.It is easy to see that for an orbit X given in uniform coordinates Xµ = xµ + Iuµ,we always have the relation u = dx

dτ . Consider an arbitrary curve Y : R −→ P1,3

whose projection π(Y ) is not everywhere timelike. Then the pseudo-imaginary partof the natural lift π(Y )∗ is not real everywhere, and Y cannot be an orbit. Hence,the projection π(X) of an orbit X is always timelike.Further observe that due to the orthogonality of dx and du = ddxdτ we have

η (dX, dX) ∈ R (59)

everywhere along an orbit X. Hence, along an orbit, the generically P-valued two-form η provides a real-valued semi inner product, allowing the following classifica-tion: An orbit X is called submaximally accelerated, if

dω2 ≡ (dX, dX) > 0 (60)

everywhere along the orbit. Note that the line element dω is an OP(1, 3) scalar.Observe that an orbit X is submaximally accelerated if and only if the projectionπ(X) has Minkowski curvature g < a. This is seen as follows. Let x ≡ π(X), u ≡ dx

dτ

and a ≡ dudτ . Then X = ax+ Iu and we have from (39)

dω2 =(1− a−2g2

)a2dτ2 (61)

where g is the Minkowski-scalar acceleration of the trajectory x. Hence,

dω2 > 0 ⇔ g < a

as dτ2 > 0 as x = π(X) is everywhere timelike.This gives us the interpretation of the constant a as the upper bound for accelera-tions in this theory, and thus justifies the terminology.

13

In analogy to the notion of rapidity in special relativity, it is useful to introduce aconvenient non-compact measure for accelerations. This will clear up notation lateron. Let X be a submaximally accelerated orbit, and g be the Minkowski curvatureof the projection π(X) for a particular uniform observer. Then the accelerity α ofthe trajectory is given by

tanh(α) =g

a. (62)

Hence the relation between the OP(1, 3)-invariant line element dω of an orbit andthe Minkowski line element dτ of the projection π(X) is

dω =a

cosh(α)dτ. (63)

Note that although dτ and α are frame dependent, the combination on the righthand side above is manifestly frame independent, as dω is.Finally, we define the eight-velocity U of a submaximally accelerated orbit X as

U : R −→ TP1,3 (64)

U ≡ dX

dω. (65)

This is well-defined due to the OP(1, 3)-invariance of dω. Sometimes we will considerthe real and pseudo-imaginary part of U in uniform coordinates, which we will denote

U ≡ u+ Ia = cosh(α)(u+ Ia−1a

), (66)

where u and a are the four-velocity and acceleration of the corresponding spacetimetrajectory π(X).

4.2 Linear Uniform Acceleration in Extended Special Relativity

It is instructive to study orbits which project to trajectories of linear uniform ac-celeration, i.e. constant accelerity α. Let U be the eight-velocity of such an orbit.From η(U,U) = 1 and using (62), (66), we get

uµuµ = 1 + sinh2(α), (67)uµaµ = 0. (68)

Choosing a Lorentz frame such that

u2 = u3 = a2 = a3 = 0,

(68) becomes

u0a0 − u1a1 = 0, (69)

14

which is solved by

a0 = γu1, (70)a1 = γu0, (71)

for some function γ. Constant accelerity gives

aµaµ = − cosh2(α)g2

a2= − sinh2(α).

On the other hand,

aµaµ = −γ2uµuµ = −γ2(1 + sinh2(α)

).

Hence for linear uniform acceleration of modulus g,

a0 =g

au1, (72)

a1 =g

au0, where g < a. (73)

Now consider the projections

π01(u0, u1, a0, a1) = (u0, a1), (74)π10(u0, u1, a0, a1) = (u1, a0) (75)

from TP1,3 to the u0 − a1 and u1 − a0 planes, respectively (see figure 1). From (72)

and (73), we see that an orbit corresponding to a spacetime trajectory of constantMinkowski curvature g projects under π01 and π10 to straight lines through theorigin of slope ag−1 in the u0 − a1 and u1 − a0 planes. Special relativity allowsarbitrarily high accelerations, hence there the spectrum of uniformly acceleratedspacetime curves is given by all straight lines through the origin in these planes,with the same slope in both planes for one particular curve.In the framework presented here, however, now g is bounded from above by a.

Using the conversion formula (66), we see that in the spacetime projection, (72-73) gives the familiar hyperbolae of Minkowski curvature g, but only for g < a.This determines the spectrum of uniformly accelerated curves in extended specialrelativity up to Poincare transformations (see figure 1).

4.3 Special Transformations in SOP(1, 3)

Exponentiation of the generators Mµν (cf. section 3.3) yields the familiar realLorentz transformations acting on TP

1,3. The group elements generated by theIMµν are given by the ordinary Lorentz transformations evaluated with purelypseudo-imaginary arguments. For notational simplicity, we exhibit their proper-ties in a (1 + 1)-dimensional theory. The pseudo-boost of accelerity β is then givenby

Λboost(Iβ) =(

cosh(β) I sinh(β)I sinh(β) cosh(β)

), (76)

15

2

t

x1

x0 2

a 1

u 02 u

a 0

1

(a) (b) (c)

1

1 1

Figure 1: spectrum of uniformly accelerated curves in MSR

and its action on the eight-velocity U = u+ Ia is

u0

u1

a0

a1

7→

cosh(β) u0 + sinh(β) a1

cosh(β) u1 + sinh(β) a0

cosh(β) a0 + sinh(β) u1

cosh(β) a1 + sinh(β) u0

. (77)

This hyperbolically rotates a straight line of hyperbolic angle β in the u0 − a1 andu1 − a0 plane to another straight line of hyperbolic angle α + β in the respectiveplane. One can check that the the two remaining pseudo-boosts do exactly the samewith the other coordinates. Hence the pseudo boosts acting on the eight-velocitymap the projections (74-75) of a U -curve to the projection of another uniformlyaccelerated curve U ′. We still have to check whether it maps the whole curve Uto a uniformly accelerated curve U ′. But this is easily seen by counting degrees offreedom: for a pseudo-boost in 1-direction, the projections of the transformed curvegive us the two constraints

a′0

= tanh(α+ β)u′1

(78)

a′1

= tanh(α+ β)u′0

(79)

and we know that

a′2

= a2 (80)

a′3

= a3 (81)

u′2

= u2 (82)

u′3

= u3 (83)

Further we know that any submaximally accelerated timelike curve lies in the hy-persurface

η(U,U) = 1

16

in U -space. Being OP(1, 3)-transformations, pseudo-boosts respect this condition,and thus it is still true for the transformed curve U ′. This leaves us with 8 − (2 +4 + 1) = 1 degrees of freedom, uniquely determining U ′.The pseudo-boosts transform the eight-velocity of a submaximally accelerated orbitin one frame to the eight-velocity in a relatively accelerated frame.

According to the scheme

x: IR Mtimelike

submax. acc.

X: IR TMorbit

(dX,dX)>0

U: IR T M2

8-velocity

8-velocity

U’: IR T M2

8-velocity(dX,dX)>0

orbitTMX’: IR

timelikesubmax. acc.

Mx’: IR

natural lift

*d

dω

dωπ

bundleprojection

induced actionon

trajectories

induced actionon

orbitson

linear action

this induces an action of the pseudo-boosts on the orbit X, as with given initialconditions Xinitial, X is uniquely determined by U . Note that this only means thatthere is an action of the pseudo-boosts induced on the space of motions, but not onspacetime. Indeed, in section 4.5, we find that in the spacetime projection, this givesrise to a non-commutative geometry. In particular, there can be no point mappingR

1,3 −→ R1,3 describing the action of pseudo-boosts on spacetime.

Pseudo-boosts in an arbitrary space direction can be composed from the pseudo-boosts in the coordinate directions and an appropriate rotation, exactly like for realboosts:

rotation−1R boostn rotationR = boostRn,

rotation−1R p-boostn rotationR = p-boostRn.

The role played by the pseudo-rotations is illuminated by the identity

p-rot−1i

(π2

)boosti p-roti

(π2

)= p-boosti.

The pseudo-rotations rotate velocities into accelerations and vice versa, thus showingexplicitly that there is no well-defined distinction between velocities and accelera-tions, like in canonical classical mechanics due to symplectic symmetry. We willsee this mechanism at work explicitly when discussing point particle dynamics inchapter 7.

17

4.4 Physical Postulates of Extended Special Relativity

Equipped with the machinery developed above, we can now concisely formulate thephysical postulates of the maximal acceleration extension of special relativity.

Postulate I.Massive particles are described by submaximally accelerated orbits X, i.e.

η (dX, dX) > 0 (84)

everywhere along X.

Postulate II. (modified clock postulate, [9])The physical time measured by a clock4 with submaximally accelerated orbit X isgiven by the integral over the line element,

Ω = a−1

∫dω = a−1

∫ √η

(dX

dλ,dX

dλ

)dλ. (85)

For curves of uniform accelerity α, the modified clock postulate gives a departurefrom the special relativity prediction by a factor of

cosh−1(α) =

√1− g2

a2. (86)

Hence experiments testing the clock postulate and involving accelerations |g| <gexp give a lower bound on the hypothetical maximal acceleration a.Farley et al. [10] have measured the decay rate of muons with acceleration gexp =5× 1018ms−2 within an accuracy of 2 percent. This corresponds to a measurementof the lifetime within the same accuracy ∆ = 0.02. Hence the factor (86) mustdeviate from unity less than ∆: √

1− g2

a2> 1−∆, (87)

thus leading to an experimental lower bound for the maximal acceleration

a >g√

∆ (2−∆)≈ 2.5 × 1029ms−2 (88)

Correspondence PrincipleFrom the postulates above, we recognize that in the limit a →∞, we have dω → dτ ,and U → u, i.e. extended special relativity becomes special relativity.

4e.g. an atomic clock

18

4.5 Born-Infeld Theory Revisited

After the formal developments in the last two chapters, we return to the start-ing point of our investigations, the Born-Infeld action. We demonstrate that thetransformation of the commutation relations (12) as a second rank tensor5 of thepseudo-complex Lorentz group is well-defined, and hence Born-Infeld theory is com-patible with the extended specially relativistic kinematics developed earlier in thischapter. This shows that the pseudo-complexification procedure was indeed success-ful in the kinematization of the Born-Infeld symmetries associated with the maximalacceleration.

In section 2, we assumed a particular coordinate non-commutativity in order torecast the Born-Infeld action in the suggestive form (16). Now we are in a positionto prove that this is is the only well-defined non-commutative geometry admittedby SOP(1, 3) symmetry. Assume the commutation relations in the background ofan electromagnetic field are more generally given by

[xµ, xν ] = −ieKµν , (89)[xµ, pν ] = −iηµν , (90)[pµ, pν ] = ieFµν , (91)

with a (so far) arbitrary, antisymmetric Kµν . Using the notation Xm = (axµ, uµ),we define the tensor H through

[Xm,Xn] = −i a

m

( −eamKµν −ηµνηµν − e

amFµν

)︸ ︷︷ ︸

≡Hmn

(92)

and require it transforms as a second rank tensor under SOP(1, 3):

Hmn −→ SmaHabSnb = SmaH

ab(St

)b

n. (93)

The transformation rule (93) is consistent with what we expect from the action ofreal Lorentz-transformations on F , K and g:(

K g

−g F

)−→

(ΛKΛt ΛgΛt

−ΛgΛt ΛFΛt

). (94)

Now we calculate the action on H of a pseudo-boost with accelerity β in spatial1-direction. Consider the special case of F = K = 0 and g = η, i.e.

Hab =(

0 ηαβ−ηαβ 0

). (95)

For calculational simplicity, we only consider the (1 + 1)-dimensional case and weget from (95) using the transformation law (93):

Hab −→

0 2 sh ch sh2 + ch2 0−2 sh ch 0 0 −(sh2 + ch2)

−(sh2 + ch2) 0 0 −2 sh ch0 sh2 + ch2 2 sh ch 0

, (96)

5We stick to the term tensor (vetcor, scalar) although they are defined merely on a module andnot on a vectorspace. 19

where ch = cosh(β), sh = sinh(β) and β > 0, i.e. S is a prop in the negative (!) x1

direction.We observe two related points

1. the SOP(1, 3)-transformation preserves the antisymmetry of H, which is ofcourse crucial for the interpretation of the component blocks K, F and g, andin turn for the well-definition of H.

2. the mixing behaviour shows that, for reasons of consistency, it is necessary toidentify the mixed parts of K and F up to constant factors:

K0i =−1

m02a2

F 0i i = 1, 2, 3 . (97)

Hence we recognise that either the presence of an electromagnetic field or the changeto a relatively accelerated frame introduces a (a−2-suppressed) time-position non-commutativity, of exactly the form required to give (16).It is easy to see that the expression

det (H) (98)

is invariant under the SOP(1, 3)-transformation (93). Hence the Born-Infeld La-grangian

LBI =√

det (gµν + bFµν) (99)

can be written as the manifestly SOP(1, 3)-invariant expression

LBI = det (H)14 (100)

for a distinguished choice of the Born-Infeld parameter, i.e. b = |e|m−1a−1, so thatrelation (7) now follows from pseudo-complex Lorentz invariance!

5 Pseudo-Complex Manifolds

Our findings in the flat case R1,3 motivate a generalization to a generally curved

n-dimensional manifold M . In particular, the isomorphism (43)

(Pn, η) ∼= (TR

n, ηD, ηH), (101)

suggests to consider the tangent bundle TM , equipped with two metrics of signa-ture (2, 2n − 2) and (n, n), respectively. Remarkably, much of such a mathematicalframework has already been developed by Yano and others [8] from a pure math-ematical point of view, which we can now give a physical interpretation from ourinsights gained in the flat case.

20

5.1 Lifts to the Tangent Bundle

Several kinds of lifts of geometrical objects from a base manifold M to its tangentbundle TM are introduced, and some of their properties essential for our purposesare explored. We use local coordinates for all our definitions, but everything can bemade coordinate-free as shown in e.g. [8].

Throughout this chapter, M denotes a differentiable manifold, g a metric and ∇ alinear connection on M . π : TM −→ M denotes the canonical bundle projection.Let xµ be local coordinates on M . The induced coordinates for a point (x, y =yµ∂µ) ∈ TM are (xµ, yµ). The shorthand notations

∂µ ≡ ∂

∂xµ∂µ ≡ ∂

∂yµdxµ ≡ dyµ (102)

are useful to clear up notation.We now define the action of vertical and horizontal lifts of functions, vectors andone-forms, and then algebraically extend these definitions to tensors of arbitrarytype [8].

Vertical Lifts are defined on any differentiable manifold.

1. fV ≡ f π2. (∂µ)

V ≡ ∂µ

3. (dxµ)V ≡ dxµ

4. algebraic extension via(P +Q)V = P V +QV

(P ⊗Q)V = P V ⊗QV

Horizontal Lifts are defined on manifolds carrying a linear connection ∇ withChristoffel symbols Γσµτ .

1. fH ≡ 0

2. (∂µ)H ≡ ∂µ − Γτ µ∂τ

3. (dxµ)H ≡ dxµ + Γµτdxτ

4. algebraic extension via(P +Q)H = PH +QH

(P ⊗Q)H = PH ⊗QV + P V ⊗QH

where Γµτ ≡ yσΓσµτ .A third type of lifts to the tangent bundle, diagonal lifts, are only defined for (0, 2)tensors on a manifold with symmetric affine connection:

21

Diagonal Lift Let G be a (0, 2) tensor on (M,Γ)

GD ≡ (Gµν)V (dxµ)V ⊗ (dxν)V + (Gµν)

V (dxν)H ⊗ (dxµ)H (103)

As in the flat case, the natural lift x∗ of a curve x : R −→ M plays an importantrole. It is given in induced coordinates by

X : R −→ TM

Xm ≡(

axµ,dxµ

dτ

), (104)

where dτ is the arc length of the curve x on (M,g).

5.2 Adapted Frames

The induced frame on TM ,

∂µ, ∂µ (105)

allows an easy interpretation of a tangent bundle vector, but for many calculationalpurposes, the so-called adapted frame is more convenient. The 2n local vector fields

e(µ) ≡ (∂µ)H , e(µ) ≡ (∂µ)

V (106)

constitute a basis of the tangent space TX(TM) of the tangent bundle at pointX ∈ TM , the so-called adapted frame. The components of these basis vectors ininduced coordinates are

e(µ) =(

δνµ−Γµν

), e(µ) =

(0δνµ

). (107)

The dual coframe is

εµ ≡ (dxµ)H , εµ ≡ (dxµ)V . (108)

In the adapted frame, horizontal and vertical lifts are similarly easy to perform.Consider the vector X = Xµ∂µ and the one-form ω = ωµdx

µ. In the adapted frame,

XH =(Xµ

0

), XV =

(0Xµ

)(109)

ωH = (0, ωµ), ωV = (ωµ, 0). (110)

The transformation between induced and adapted coordinates has unit determinant,as follows from [8].

22

5.3 Lifts of the Spacetime Metric

The above definitions allow to calculate the component form of the horizontal anddiagonal lifts of the metric g of a semi-Riemannian manifold (M,g) with the con-nection being the Levi-Civita connection.In induced coordinates,

gD =(gij + gtsΓtiΓsj Γtjgti

Γtigtj gij

), (111)

gH =(∂gij gijgji 0

), (112)

where ∂ ≡ yi∂i.Using the adapted frame, this becomes

gD =(gµν 00 gµν

)= gµν ⊗ 1, (113)

gH =(

0 gµνgµν 0

)= gµν ⊗ I. (114)

One can check that for a metric g of signature (p, q) on M , both gD and gH areglobally defined, non-degenerate metrics on TM with signature (2p, 2q) and (p +q, p+ q), respectively. Observe that in the flat case g ≡ η, we have

gD = η ⊗ 1, (115)gH = η ⊗ I, (116)

also in the induced frame, which we used throughout chapter 4.

5.4 Connections on the Tangent Bundle

Let ∇ be the (torsion free) Levi-Civita connection on TM with respect to gH , i.e.

∇gH = 0. (117)

Denote the Christoffel symbols Γ. In terms of the Christoffel symbols Γ formed withthe metric g on M , we find

Γ τµ ν = Γ τ

µ ν ,

Γ τµ ν = ∂Γ τ

µ ν ,

Γ τµ ν = Γ τ

µ ν , (118)

Γ τµ ν = Γ τ

µ ν ,

all unrelated symbols of Γ being zero.The serious drawback of this connection is that in general

∇gD 6= 0. (119)

23

At the expense of introducing torsion on the tangent bundle, however, we can finda connection ∇H which makes both metrics simultaneously covariantly constant:

∇HgH = 0, (120)∇HgD = 0. (121)

Denoting its Christoffel symbols Γ, they are in terms of the Christoffel symbols Γ:

Γ τµ ν = Γ τ

µ ν ,

Γ τµ ν = ∂Γ τ

µ ν −Rσµντyσ,

Γ τµ ν = Γ τ

µ ν , (122)

Γ τµ ν = Γ τ

µ ν ,

all unrelated components of Γ being zero, and R is the Riemann curvature of Γ.The torsion on TM is then

2Γ τ[ν µ] = Rµνσ

τyσ (all unrelated vanishing) (123)

in induced coordinates.The connection ∇H has the further nice properties

1. ∇H has vanishing Riemann tensor iff ∇ has vanishing Riemann tensor. [11]

2. ∇H has vanishing Ricci tensor iff ∇ has vanishing Ricci tensor. ([8], IV.4.3)

3. ∇H has gD-Ricci scalar R if ∇ has Ricci scalar R (see section 6.2).

4. ∇H has vanishing gH -Ricci scalar. ([8], IV.4.4)

We will make use of these curvature properties when lifting the Einstein field equa-tions in section 6.2.

5.5 Orbidesics

Exactly as in the flat case, a curve X : R −→ TM is called an orbit, if

X = π(X)∗, (124)

i.e. if the natural lift of its bundle projection recovers the curve. Again, in inducedcoordinates Xm = (xµ, yµ), this is equivalent to yµ = dxµ

dτ , where τ is the arc lengthof the curve x in (M,g).As we are dealing with real manifolds, the orbital condition (124) also ensures thatthe projection π(X) is always timelike. However, in the general curved case, thedistinguished role of the orbits will be seen of much wider importance. We thereforeintroduce the notions of orbidesics and orbiparallels. An orbit X which is also anautoparallel with respect to some connection ∇ is called a ∇-orbiparallel. If, morespecially, the connection is the Levi-Civita connection of some metric G, then X iscalled a G-orbidesic.The following statements follow from ([8],I.9.1 and I.9.2)

24

1. Let X be a gH-orbidesic on TM . Then π(X) is a geodesic on (M,g).

2. Let x be a geodesic on (M,g). Then x∗ is a gH -orbidesic and a gD-orbidesic.

A direct corollary is that any gH -orbidesic is a gD-orbidesic, but the converse doesnot hold.Diagrammatically,

@@

@@@R@

@@

@@I

-gH-orbidesic on TM gD-orbidesic on TM

g-geodesic on M

* *

π

The gD-line element of an orbit X is denoted dω and given by

dω2 = gD (dX, dX) . (125)

The spacetime line element of a bundle projection of an orbit X is given by

dτ2 = g(dπ(X), dπ(X)). (126)

An important observation are the relations

gH(uH , uH) = 0, ([8],p. 140) (127)gD(uH , vH) = [g(u, v)]V , ([8],IV.5.1) (128)

for vector fields u, v on M . These relations allow to recognize that the unit tangentvector to a gH -orbidesic X is just the horizontal lift of the four-velocity of π(X),

dX

dω=

(dx

dτ

)H

. (129)

This can be seen as follows.(dx

dτ

)H

=(dxµ

dτ,−dx

α

dτΓαµβ

dxβ

dτ

)=dX

dτ(130)

using the geodesic property of x = π(X). Hence,

dω2 = gD(dX

dτ,dX

dτ

)dτ2 = g

(dx

dτ,dx

dτ

)dτ2 = dτ2 (131)

using the second relation above. Thus,

gD (dX, dX) > 0 for any gH − orbidesic. (132)

For orbits which are not gH -orbidesical, this is neither generally true nor generallyfalse. This allows the following classification: An orbit X is called submaximallyaccelerated, if gD (dX, dX) > 0 everywhere along the orbit. The result (132) isreassuring, as orbidescial motion will represent unaccelerated motion.

25

5.6 Orbidesic Equivalence

We found that there is a one-to-one correspondence between g-geodesics and gH -orbidesics. However, in the next section we will explain why ∇H rather than ∇ isthe appropriate connection on TM , and it is desirable to learn about the relationbetween g-geodesics and ∇H-orbiparallels. The remarkable observation of this shortsection will be that the gH -orbidesics are the ∇H-orbiparallels, and vice versa.The following statements are shown in ([8], II.9.1 and II.9.2).

1. Let X be an ∇H-orbiparallel. Then π(X) is a g-geodesic.

2. Let x be a g-geodesic. Then x∗ is a ∇H-orbiparallel.

This completes the diagram from the last section to

@@

@@@R@

@@

@@I

6

?

-gH-orbidesic on TM gD-orbidesic on TM

g-geodesic on M

* *

π

∇H -orbiparallel on TM

*

π

5.7 The Tachibana-Okumura No-Go Theorem

In 1981, Caianiello observed that requiring positivity of tangent bundle curves withrespect to the metric (

η 00 η

), (133)

which generalizes to gD in the curved case, introduces an upper bound on worldlineaccelerations [12]. Given the importance of the complex structure of the phase spacein Hamiltonian mechanics, it seemed quite sensible to establish a complex structurealso on the tangent or cotangent bundle of Minkowski spacetime, and several at-tempts have been made in both the flat [7] and the curved case [6].

We are now in a position to prove that a complex structure F on the tangent bundleis incompatible with the assumption of a maximal acceleration introduced by gD inthe sense explained above. This statement is true under the physical assumption thata strong principle of equivalence between the flat and the curved case holds, or in

26

mathematical terms, that the structures gD and F are required to be simultaneouslycovariantly constant:

∇gD != 0, (134)

∇F != 0. (135)

In this approach, gD is the only metric on TM , and so we take ∇ to be the Levi-Civita connection with respect to gD. The globally defined one-form on TM

Θ ≡ −gµνyµdxν (136)

has an exterior differential

dΘ =12FmndX

mdXn. (137)

Raising and lowering indices with gD, we can easily verify that

FamFn

a = −δmn , (138)

hence Fmn defines an almost complex structure on TM . The covariant derivativeof Fmn with respect to ∇ evaluates to [8]

∇µFνα = −∇µFν

α =12

(Rαµσν +Rµνσα) yσ, (139)

∇µFνα = ∇µFν

α =12Rανσµy

σ, (140)

and all other terms vanish. This immediately gives the

Tachibana-Okumura No-Go Theorem [13]The tangent bundle TM of a semi-Riemannian manifold M has simultaneously co-variantly constant metric gD and complex structure F if and only if the base manifoldM is flat.

This makes all past approaches in this direction physically questionable. As pointedout in section 2, even though for flat Minkowski spacetime, equipping the tangentbundle with a complex structure is compatible with a maximal acceleration, theslightest perturbation would render the theory inconsistent, and a generalization tothe curved case is entirely frustrated.

Clearly, the pseudo-complex approach presented in this paper circumvents the Tachi-bana-Okumura no-go theorem.

27

6 Maximal Acceleration Extension of General Relativ-

ity

The stage of extended general relativity is the tangent bundle TM of curved space-time M , equipped with the horizontal and diagonal lift of the spacetime metricg,

(TM, gH , gD

). As the linear connection we take ∇H . Then both metrics are

covariantly constant:

∇H gH = 0, (141)∇H gD = 0, (142)

and we know that∇H has the same orbidesics as ∇, the Levi-Civita connection of gH .Hence we can establish the strong principle of equivalence within this framework,circumventing the Tachibana-Okumura no-go-theorem.

6.1 Physical Postulates of Extended General Relativity

We require for all orbits representing submaximally accelerated particle motion that

1. gH(dX, dX) = 0,

2. gD(dX, dX) > 0.

For orbidesics, we saw these conditions are automatically satisfied.

We measure physical time along an orbit X by

dω =[gD(dX, dX)

]− 12

Particles, under the influence of gravity only, travel along (gH -null) gH -orbidesics,or equivalently, ∇H-orbiparallels.

Hence for particles in a gravitational field only, the orbits are also gD-geodesics,and their proper time is measured with gD as well. One could say that we onlyreally need the two metrics gH and gD if it comes to non-geodesic orbits. This canbe seen as a handwaving explanation, why in general relativity on spacetime withsmall (non-gravitational) acceleration, one metric always seemed to be enough.In the absence of any force besides gravity, extended general relativity is exactlyequivalent to general relativity, as desired.

6.2 Field Equations

In order to achieve a formulation of extended general relativity entirely in terms oftangent bundle concepts, it is certainly necessary to lift the field equations for g on

28

spacetime to field equations for gH and gD on the tangent bundle.The Ricci tensor R of the connection ∇H evaluates to

Rab =(Rαβ 00 0

), (143)

in induced and adapted coordinates alike. We recognize that

Rab = (Rαβ)V . (144)

Unlike the Ricci tensor, the Ricci scalar depends on the metric used for its contrac-tion,

Rab(gH

)ab= 0, (145)

Rab(gD

)ab= R, (146)

where R ≡ Rαβgαβ is the curvature scalar on M . This can be immediately seen in

the adapted frame. Thus it is sensible to define the Ricci scalar R on TM as

R ≡ Rab(gD

)ab. (147)

Now consider the Einstein field equations on M ,

Rαβ − 12gαβR = 8πGTαβ . (148)

’Duplication’ trivially gives an equivalent set of equations(Rαβ 00 Rαβ

)− 1

2

(gαβ 00 gαβ

)R =

(Tαβ 00 Tαβ

)(149)

Observe that the first term on the left can be interpreted in the adapted frame as[(gD

)am (gD

)bn+

(gH

)am (gH

)bn]Rmn. (150)

The second term equals

−12

(gD

)ab (gD

)mn+

(gH

)ab (gH

)mn︸ ︷︷ ︸=0

Rmn. (151)

Defining the ’double metric’

Gabcd ≡ (gD

)ab (gD

)cd+

(gH

)ab (gH

)cd, (152)

we find from (149) the tangent bundle tensor equation(Gambn − 1

2Gabmn

)Rmn = T ab, (153)

where

T ab ≡(Tαβ

)D. (154)

The equations (153) are fully equivalent to the Einstein field equations (148), andwe call them the lifted field equations. Being a tensor equation, (153) is valid in anyframe, not just the adapted frame we used for its derivation.

29

7 Point Particle Dynamics

7.1 Free Massive Particles

In a series of carefully written papers, Nesterenko et al. [1, 14, 15] investigate intothe Lorentz invariant action

S =∫ √

a2 − g2dτ, where dτ2 = dxµdxµ, g2 = −d2xµ

dτ2

d2xµdτ2

(155)

within the framework of special relativity. As the associated Lagrangian dependson first and second order derivatives,

L = L (xµ, xµ) , ˙≡ d

dt, (156)

the Euler-Lagrange equations are

d2

dt2∂L

∂xµ− d

dt

∂L

∂xµ+

∂L

∂xµ= 0, (157)

and transition to the Hamiltonian formalism is more complicated as for first orderLagrangians, though possible. It is shown in [14, 15] that the action (155) providesviable specially relativistic dynamics for a free massive particle, consistent with theassumption of an upper bound a on accelerations.Nevertheless, the appearance of Lagrangians with second order derivatives is a con-siderable inconvenience, and makes the theory look prone to encountering difficultiesat a later stage, e.g. quantization, even if it can be shown to be consistent in par-ticular cases as above.However, the second order derivatives in (155) were introduced in order to dynami-cally enforce a maximal acceleration in the framework of special relativity. Writingthe action in the manifestly OP(1, 3)-invariant form

S =∫dω =

∫ √XµXµdt, Xµ ≡ xµ + Iuµ, (158)

and, sloppily speaking, leaving the rest to the extended relativistic kinematics,’miraculously’ solves the problems mentioned above: the relation between the four-velocity and four-acceleration is absorbed in the pseudo-complex tangent bundlegeometry, and hence (158) is not just a notational trick. Moreover, the pseudo-complexification prescription

R1,3 −→ P

1,3 (159)

turns out to be equally applicable to Lagrangians, in the present case converting theaction of a free relativistic point particle to the extended relativistic action (158),automatically generating the necessary constraints!Start with a specially relativistic point particle of mass m,

S = m

∫ √xµxµdt. (160)

30

It is convenient to rewrite the Lagrangian in ’Hamiltonian form’, explicitly includingthe constraint on the associated canonical momenta:

L = pµxµ − 1

2λ

(pµp

µ −m2). (161)

We apply the pseudo-complexification prescription (159) and replace

xµ −→ Xµ ≡ xµ + Iuµ, (162)pµ −→ Pµ ≡ pµ + Ifµ, (163)

obtaining

LMSR = PµXµ − 1

2λ

(PµP

µ −m2). (164)

Note that naively performing derivatives

∂

∂Xµ

∂

∂Xµ(165)

will lead into trouble, as P is only a ring, and hence the differential quotient is notgenerally defined. However, from the fully equivalent tangent bundle point of view,the definition

∂

∂Xµ≡ 1

2

(∂

∂xµ+ I

∂

∂uµ

)(166)

is easily understood, and turns out to be a useful one. Thus we see that

∂L

∂Xµ= pµ + Ifµ = Pµ (167)

really gives Pµ as the canonical momentum conjugate to Xµ, as suggested by thenotation.We obtain the equations of motion from (164) by variation with respect to e, P , andX:

PµPµ = m2, (168)

Xµ − λPµ = 0, (169)d

dtPµ = 0. (170)

Relations (169) and (170) immediately give

Xµ = 0, (171)

and from (168) and (169)

XµXµ = λ2PµPµ = λ2m2 > 0, (172)

31

hence the particle is submaximally accelerated! We choose the gauge λ = m−1,corresponding to natural parameterization ω, with dω2 = dXµdXµ. Then from(171),

Xµ = cµ + Idµ, (173)

for constant real four-vectors c, d ∈ R1,3 satisfying

cµcµ + dµdµ = 1 (174)cµdµ = 0. (175)

We remark in passing that (175) means we are looking for ηH-null geodesics (cf.section 6.1) Note that this set of conditions is OP(1, 3) invariant. These conditionsenforce that exactly one of c and d must be timelike, and the other one spacelike orvanishing. As the equations (174-175) are invariant under exchange of c and d, wecan assume without loss of generality that c be timelike. If then d is vanishing, weget the solution

Xµ = cµ, c unit timelike. (176)

If d is spacelike, we can (due to the OP(1, 3)-invariance of the conditions) performa boost such as to get c = (γ, 0, 0, 0), and thus because of (175), d = (0, δ1, δ2, δ3),which we can rotate to d = (0, δ, 0, 0) without changing c. Then the pseudo-rotation

1 0 0 00 0 −I 00 I 0 00 0 0 1

(177)

will also give a solution of the form (176). Using the pseudo-complex Lorentz symme-try of the equations, we find from (176) all other solutions satisfying the contraints:

Xµ = cµ + Idµ. (178)

This may look strange, but is easily understood as a consequence of the pseudo-complex Lorentz invariance of the theory, as the transfomations

OP(1, 3)\OR(1, 3) (179)

are the analoga of symplectic transformations in classical canonical mechanics, wheresymplectic symmetry also does not allow for a well-defined distinction between co-ordinates and momenta!

It is interesting to note that pseudo-complexification of the unconstrained relativisticLagrangian yields

LSR ≡ xµxµ −→ LMSR ≡ XµXµ = (xµxµ + uµuµ) + 2Ixµuµ, (180)

allowing us to enforce the orthogonality constraint by a reality condition

LMSR!= LMSR. (181)

32

7.2 Kaluza-Klein induced coupling to Born-Infeld theory

The discussion of a free submaximally accelerated particle in the previous sectionseems somewhat academic, as if there is no external force present, the particle is triv-ially submaximally accelerated, as it is moving along a geodesic. Still, the exercisewas worthwile as we obtained a manifestly OP(1, 3)-invariant first order Lagrangianfor a free massive particle. Born-Infeld electrodynamics (2), having sparked thewhole investigation, provides a suitable candidate for an extended specially rela-tivistic external force. We now set out to construct an interaction term, coupling amassive particle to Born-Infeld electrodynamics. Conventional minimal coupling

Lm.c. = −exµAµ, (182)

as provisionally assumed in (4), does not do the job, as it is not OP(1, 3)-invariant.It is well-known that for a specially relativistic point particle, coupling to the elec-tromagnetic field can be achieved using the Kaluza-Klein approach

L = − (z −Aµxµ)2 + gµν x

µxν , (183)

where (xµ, z) ∈ R1,3 × S1, and the conserved momentum conjugate to the cyclic

variable z,

e ≡ ∂L

∂z= −2 (z −Aµx

µ) (184)

is interpreted as the electric charge of the particle, as (183) leads to the equationsof motion

xµ + Γµαβxαxβ = eFµαxα, (185)

giving the Lorentz force law.In case of small electromagnetic fields, (183) can be written

L =(gµν AµAν −1

)mn

dxmdxn ≡ gmndxmdxn, (186)

where xm ≡ (xµ, z), and the dilaton was set to −1 (as the extra dimension must bespatial for reasons of causality). When coupling to Maxwell theory, the restrictionto small electromagnetic fields is quite problematic due to the well-known field sin-gularities. With Born-Infeld theory providing the external force, on the other hand,we are much better off in this respect. For flat background spacetime geometry, itis therefore tempting to simply pseudo-complexify the relativistic Lagrangian (183),as this was so successful in obtaining the extended relativistic version of the freeparticle. But the fact that even for flat spacetime g = η the Kaluza-Klein manifoldis curved, is a clear enough caveat and we choose to be careful and use the fullmachinery of the generally curved case. Remarkably, it will turn out that for slowlyvarying vector potential A, pseudo-complexification would have done the job equallywell.

33

In order to facilitate interpretation of the following results, we take the burden tocalculate the diagonal lift of the Kaluza-Klein metric in induced coordinates, i.e.

(gmn)D =

(gmn + gtsΓtmΓsn Γtngtm

Γtmgtn gmn

). (187)

The (4 + 1)-dimensional Levi-Civita connection evaluates for flat spacetime g = ηto

Γaβγ =12ga4Dβγ , (188)

Γa4γ =12gaµFµγ , (189)

Γa44 = 0, (190)

where

Fµν ≡ Aµ,ν −Aν,µ, (191)Dµν ≡ Aµ,ν +Aν,µ. (192)

Denote points of the tangent bundle of the Kaluza-Klein manifold

XM ≡ (xm, um) ≡ ((xµ, z), (uµ, w)) . (193)

Defining

Eφrs ≡ Dνφδνr δ

4s + Fsφδ

4r , F4π ≡ 0 ≡ F44, (194)

we obtain

gstΓtφ =12Eφrsu

r, (195)

gstΓt4 =12Fsνu

ν , (196)

determining the off diagonal blocks of (187). Further, one finds

[gstΓtiΓsj

]i=φj=ψ

=14grpEφsrEψtpu

sut, (197)

[gstΓtiΓsj

]i=φj=4

=14grtEφsrFtνu

suν , (198)

[gstΓtiΓsj

]i=4j=ψ

=14grsEψtrFsνu

tuν , (199)

[gstΓtiΓsj

]i=4j=4

=14gσµFσνFµρu

νuρ, (200)

34

determining the left upper block in (187). Observing that Eφrψ = −Eψrφ, we get

L =(gD

)MN

dXMdXN (201)

= gij(xixj + uiuj

)(202)

+12

(Dνφu

ν xφw + Fφνuν zuφ

)(203)

+22

(Dνψu

ν uψ z + Fψνuνwxψ

)(204)

+14

(grpEφsrEψtpu

sutxφxψ + gσµFσνFµρuνurhoz2

)(205)

+12gtrFtνEφsru

νusxφz. (206)

For slowly varying vector potential A, all terms D, F , and hence E are small, andto lowest order, we get

L0 = − (z −Aµxµ)2 − (w −Aµu

µ)2 + ηµν (xµxν + uµuν) , (207)

the Lagrangian for a free extended specially relativistic particle, plus an OP(1, 3)-invariant coupling term! This result justifies to push the analysis for slowly varyingvector potentials A a bit further: We will redo the calculation, now using the pseudo-complexification prescription to obtain the extension of (183), thus being able to takecare of constraints by imposing the reality condition (181).Direct pseudo-complexification

xµ −→ xµ + Iuµ, (208)z −→ z + Iw, (209)

of the Lagrangian (183) yields

Lconstr0 = L0 + I − (z −Aµx

µ) (w −Aµuµ) + gµν x

µuν , (210)

generating an additional pseudo-imaginary part compared with (207), which we willcome back to in a moment.The conserved quantities

e1 ≡ −2 (z −Aµxµ) , (211)

e2 ≡ −2 (w −Aµuµ) , (212)

are easily interpreted from the equations of motion for (210),

Xµ = (e1 + Ie2)Fµν (xµ + Iuµ) . (213)

Additional to the familiar coupling of the velocity to the electromagnetic field, con-trolled by e1, there is now also an a priori possible coupling of the acceleration,controlled by e2. However, if we now impose the reality condition on (210),

L!= L, (214)

35

in order to generate the constraints, we find

gµν xµuν =

e1e24. (215)

In contrast, the orthogonality condition

gH(X, X

)= 0 (216)

for any orbits representing submaximally accelerated particles shows that for elec-trically charged particles, we must set e2 ≡ 0, i.e. there is no such ’accelerationcoupling’ possible in the framework of extended relativity. So from (213) we ob-tain the equation of motion for a submaximally accelerated particle coupled to a(Born-Infeld) electromagnetic field as

Xµ = eFµνXν , (e being the electric charge) (217)

which, of course, is roughly speaking just two ’copies’ of the Lorentz force law inthe real and pseudo-imaginary part. This leads to the remarkable conclusion thatthe Lorentz force law as such is also extended specially relativistic, without anymodification!

8 Quantisation

Classical Hamiltonian mechanics is the study of (non-relativistic) phase space func-tions and their evolution in time determined by the Hamiltonian H of the systemat hand. Classical phase space carries a complex structure ωij , satisfying

ωajωjb = −δab . (218)

It is well-known that defining the Poisson bracket

[F,G]P.B. ≡ ωij∂iF ∂jG, (219)

where indices run over all phase space axes, the time evolution of a classical observ-able F = F (X) is determined by

[H,F ] =dF

dt. (220)

In particular, the classical Poisson bracket relations for the phase space coordinatesfollow from Hamilton’s equations as

[qm, qn]P.B. = 0, (221)[qm, pn]P.B. = δmn , (222)[pm, pn]P.B. = 0. (223)

36

Wigner’s prescription for the transition to quantum mechanics simply consists of aone-parameter (~) deformation of the classical Poisson bracket. Notably, it does notinvolve promotion of classical phase space functions to operators acting on a Hilbertspace, but is nevertheless a fully equivalent description, as shown in e.g. [16].Defining the star product

∗ ≡ exp(i~

2

←∂i ω

ij→∂j

), (224)

the Moyal bracket

[F,G]M.B. ≡1i~

(F ∗G−G ∗ F ) (225)

provides the desired deformation, as can be seen from its expansion in ~:

[F,G]M.B. = [F,G]P.B. +O(~). (226)

Note that all contributions from even powers of the star product (224) cancel in thedefinition of the Moyal bracket (225). In the Wigner formalism, the anti-symmetricMoyal bracket plays a role analogous to the commutation relations in the operatorformalism. Hence the commutation relations in the quantum theory stem from thecomplex structure of classical phase space.Now consider extended special relativity and the associated phase space(

P4, η

) ∼= (TR

4, ηH , ηD), (227)

featuring the pseudo-complex structure ηH , satisfying(ηH

)am

(ηH

)mb= +δab . (228)

In view of what we found above, this apparently does not give rise to commutationrelations. However, define the moon product

≡ sinh(i~

2

←∂i

(nH

)ij →∂j

), (229)

dropping all even powers compared to the star product (224). This enables us todefine an anti-Moyal bracket

F,GM.B. ≡1i~

(F G+G F ) , (230)

whose expansion in ~ yields

F,GM.B. =(nH

)ij∂iF ∂jG+O(~). (231)

The lowest order term of this expansion can be interpreted as an anti-Poissonbracket. We conclude that the pseudo-complex structure of phase space in extended

37

relativity gives rise to anti-commutation relations after transition to the quantizedtheory. Spinors being more fundamental than tensors, one can construct commutingtensors from anticommuting spinors, but not the other way around.

In this sense, the pseudo-complex structure proves to embrace the structures whichare always thought of as being intimately related to a complex structure on classicalphase space.

9 Conclusion

The dynamical symmetries of Born-Infeld theory associated with the maximal ac-celeration of particles coupled to it can be encoded in a pseudo-complex geometryof the tangent bundle of spacetime.Considering the theory on this space, we classify these symmetries then as kine-matical, and indeed the corresponding symmetry group preserving the geometricalstructures contains transformations to uniformly accelerated frames and a relativis-tic analog of the classical symplectic transformations.A particularly concise prescription for the implementation of an upper bound onworldline accelerations is the pseudo-complexification of real Minkowski vector spaceto a metric module. Iteration of the pseudo-complexification process, as mathemat-ically developed in section 3.4, can be shown to put upper bounds on arbitrarilymany higher worldline derivatives beyond acceleration.The applicability of this prescription likewise to vector spaces, algebras and groupsacting on them on one hand, and merely Lorentz invariant Lagrangians on the other,in order to translate them to their extended relativistic counterparts, makes the for-malism so worthwile.In the generally curved case, the pseudo-complex structure surfaces again manifestlyin the adapted frames. For the purposes of this essay, however, we made use of thenumerous results from differential geometry of tangent bundles and thus illustratedthe bimetric real manifold point of view.The lift of the Einstein field equations, and notably the recasting of Lagrangianswith second order derivatives (’dynamically enforcing’ maximal acceleration) intofirst order form, were only possible because of the identification and use of thepseudo-complex phase spacetime structure.The lifted Kaluza-Klein mechanism proved successful in generating an extendedspecially relativistic coupling of an electrically charged particle to Born-Infeld the-ory, making essential use of the relevant constructions for generally curved pseudo-complex manifolds.The pseudo-complex structure leading to anti-commutation relations in the transi-tion to the associated quantum theory sheds a new light on the ’origin’ of anticom-mutation relations, which are more fundamental than commutation relations, in thesame sense as spinors are more fundamental than tensors. These results, however,certainly deserve further investigation.

38

Acknowledgements

I would like to thank Gary Gibbons for very helpful discussions and remarks on thematerial of this paper. I have also benefitted from remarks by Paul Townsend anddiscussions with Sven Kerstan.

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